Well posedness and comparison principle for option pricing with switching liquidity

Abstract

We consider an integro-differential equation derived from a system of coupled parabolic PDE and an ODE which describes an European option pricing with liquidity shocks. We study the well-posedness and prove comparison principle for the corresponding initial value problem. * e-mails: tgulov@uni-ruse.bg (T.Gyulov) and lvalkov@uni-ruse.bg (L.Valkov)

IMEX schemes for a parabolic-ODE system of European options with liquidity shocks

“…In this section we will describe some properties of the solution to system (8) using results obtained in [4]. Also , following [6,7], two lemmas , concerning discrete maximum principle (DM) are formulated.…”

“…for some positive constants A and α. In [4], well-posedness in weighted Sobolev spaces and comparison principle for the corresponding Cauchy problem (8), (9) are established. With sufficient smoothness of the initial data the weak solutions are classical ones.…”

“…In Table 3 we list the results from computation with Scheme 2 that for this non-uniform grid the results are still first order accurate in time as in the uniform case. Now, we improve the convergence in time applying Richardson extrapolation [4]. To this aim we use the formula…”

“…where p is order of numerical solution (1 in our case) and W n is the solution obtained using time step △τ /2 and Z n is the solution obtained using time step △τ . The resulting solution Y n has order of accuracy p + 1 [4]. Table 5 shows the result of applying this technique to the Scheme 1.…”

“…The rest of the paper is organized as follows. In the next section some results concerning well-posedness of Cauchy problem for (3) and a comparison principle, obtained in [4] , are discussed . Also, two lemmas concerning discrete maximum principle [6,7] are formulated.…”